The 9 open problems.

Prove that every nontrivial zero of the Riemann zeta function $\zeta (s)$ has real part exactly $1/2$ — or exhibit a nontrivial zero off the critical line.

If $A\subseteq \mathbb{N}$ has divergent reciprocal sum ${\sum }_{a\in A}1/a=\infty$, prove it contains arbitrarily long arithmetic progressions.

For the diagonal Ramsey number $R(k)$, determine whether ${\lim }_{k}R(k{)}^{1/k}$ exists and its value. It is known that $\sqrt{2}\le \mathrm{liminf}\le \mathrm{limsup}\le 4$.

A $k$-sunflower is a family of $k$ sets with a common pairwise intersection (core). Prove that any family of more than $C_k^w$ sets of size $w$ contains a $k$-sunflower, for a constant $C_k$ depending only on $k$.

For finite $A\subseteq \mathbb{R}$, prove $\max (|A+A|,|AA|)\gg |A{|}^{2-\epsilon }$ for every $\epsilon >0$: a set cannot be both additively and multiplicatively structured.

For every fixed graph $H$, prove there is $c(H)>0$ such that every $n$-vertex graph with no induced copy of $H$ has a clique or independent set of size $\ge n^{c(H)}$.

If $A\subseteq \mathbb{N}$ is a basis of order 2 (every large integer is a sum of two elements of $A$), prove its representation count $r_A(n)$ is unbounded.

Let $\mathrm{ES}(n)$ be the least $N$ such that any $N$ points in general position contain a convex $n$-gon. Prove the conjectured exact value $\mathrm{ES}(n)=2^{n-2}+1$.

A Sidon set in $\{1,\dots ,N\}$ has all pairwise sums distinct. Its maximum size is $N^{1/2}+E(N)$; determine the true order of the error $E(N)$ (conjectured $O(N^{\epsilon })$ for every $\epsilon >0$).

Prove that every graph with minimum degree at least $3$ contains a cycle whose length is a power of $2$ — or exhibit a min-degree-$3$ graph with no such cycle.